Staff: Mentor

Such as?

(I'm just trying to clarify what exactly you're asking about, and what your "sticking points" are in your understanding, so people don't go shooting off in a half-dozen different directions, none of which may be the one you want.)

(I'm just trying to clarify what exactly you're asking about, and what your "sticking points" are in your understanding, so people don't go shooting off in a half-dozen different directions, none of which may be the one you want.)

Thanks for the reply.

My current understanding is that all quantities that are conserved must be conserved locally. If for example you had charge appearing in one place and disappearing 'instantaneously' in another place then depending on your frame of reference you would be able to see the conservation of charge violated and be able to establish whether or not you were moving.

I heard Feynman say this in one of his messenger lectures and I'm just tying to confirm whether my understanding of what he said corresponds to what he was trying to say.

I see. Well, this is a pretty difficult question to answer. The answer is that Feynman is right with his statement according to our current understanding of relativistic quantum theory, which is formulated as a socalled local, microcausal quantum field theory. This means that the Hamilton operator, describing interacting particles is built in terms of local field operators, and this implies that the conservation laws can be formulated with local continuity equations either, and in this sense all the physical laws are consistent with the local conservation of these quantities and particularly there are only local interactions, but as I said, this is so by construction of the theory, and it's done so precisely to avoid any problems which usually occur when trying to make up models which include non-local interactions. This is so, because if something in one frame of reference is non-local in the sense of spatially separated events it is necessarily non-local also in time when viewed from a different frame of reference, and then very likely problems with causality occur. That's why there is, to my, knowledge no working non-local theory found yet. At the moment, I also don't see any necessity to need one, because the local QFT description, the Standard Model of elementary particles, is so successful that physicists are desperate in trying to find a clear evidence against it to have an idea, how to go beyond this Standard Model, which for various reasons is not fully satisfactory. One reason is that it is very likely that the particle content of the Standard Model (which is complete in the sense that it describes all particles that are directly observed, and one of the largest motivations to build the LHC at CERN has been to find the one predicted particle of the Standard Model, which had not been found before, the Higgs boson, and this mission has been fulfilled in 2012, where the discovery of this last missing particle was announced) is not complete, because many astrophysical observations indicate the existence of "dark matter", i.e., matter which does not interact electromagnetically and thus doesn't radiate light. Its existence is only inferred from its gravitational interaction with visible matter and from is needed within the cosmological standard model to describe the energy-momentum content of the universe.

So, in some sense there are quite some hints that our contemporary picture of relativistic QT is not complete, but it seems unlikely that it will have to be extended in terms of some non-local theory, and the so far developed local Standard Model is in excellent agreement with all established observations, although just very recently there's some new hope to find physics beyond the Standard Model from the LHCb collaboration:

Staff: Mentor

My current understanding is that all quantities that are conserved must be conserved locally. If for example you had charge appearing in one place and disappearing 'instantaneously' in another place then depending on your frame of reference you would be able to see the conservation of charge violated and be able to establish whether or not you were moving.

I heard Feynman say this in one of his messenger lectures and I'm just tying to confirm whether my understanding of what he said corresponds to what he was trying to say.

This just implies that all global conservation laws need local equivalents.

Local conservation laws exist, and they can lead to global conservation laws - at least in special relativity, where you can still consider "the universe at this moment in my reference frame". It becomes tricky in general relativity because that concept stops being meaningful.

Sorry to repeat myself but all I'm asking is is it possible in principle for a conservation law to be non local (to have charge disappearing at one point and appearing at another point) in other words is that what Feynman said outdated.

This just implies that all global conservation laws need local equivalents.

Local conservation laws exist, and they can lead to global conservation laws - at least in special relativity, where you can still consider "the universe at this moment in my reference frame". It becomes tricky in general relativity because that concept stops being meaningful.

Is it impossible for conservation laws to be global but not local ignoring the general relativity or is ignoring general relativity an unreasonable ask?

Staff: Mentor

Sorry to repeat myself but all I'm asking is is it possible in principle for a conservation law to be non local (to have charge disappearing at one point and appearing at another point) in other words is that what Feynman said outdated.

Why do you want charge appearing and disappearing to have a conservation law?

A global conservation law and special relativity imply a local conservation law.

You are too quick for me. How do you prove that a global conservation law in SR always implies a local one? The one direction is very clear. A local conservation law is given as an equation of continuity
$$\partial_{\mu} T^{\mu \nu \rho \ldots}=0,$$
where ##T^{\mu \nu \rho\ldots}## is some tensor field. Usually one has vector fields (four-currents of conserved charges), 2nd-rank tensors (energy-momentum) and 3rd-rank tensors (angular-momentum-center-of-energy), which originate from Noether's theorem applied to global (Abelian or non-Abelian) gauge invariance, and the Poincare symmetry of local (quantum or classical) field theories.

Then the quantities
$$Q^{\nu \rho\ldots}=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} T^{0 \nu \rho \ldots}(t,\vec{x})$$
are tensors (and the volume integral over the entire space (!) is independent of the chosen reference frame due to the equation of continuity, and this is the case only if this equation of continuity holds true), and they are conserved:
$$\dot{Q}=0.$$
This can be proven with help of the four-dimensional version of Gauss's integral theorem and can be found in many textbooks like Jackson, Sexl&Urbandtke,... There you also find a discussion about useful definitions of tensors in the case, where the equation of continuity is not valid (for open systems, e.g.). Then you usually have to figure out a reference frame, somehow preferred by the physical situation (e.g., some rest frame is there is one for a given configuration like a charged capacitor). Then you can define covariant quantities by referring always to this preferred frame and write the corresponding volume integral in this frame in terms of a covariant hypersurface integral in Minkowski space (see Jackson for a careful treatment of this point), but that's not the point in this posting.

Now the question is: Is this line of arguments reversible, i.e., can I start from the integral ##Q^{\nu \rho \ldots}## and the assumption that it is a constant in time to prove that (a) ##Q^{\nu \rho\ldots}## is a tensor and (b) that it can be written as an integral of a tensor (!) field, for which the corresponding equation of continuity (i.e., the local conservation law) holds? I've the feeling that's a pretty strong theorem, if valid at all, because in the arguments above, you have to take the integral over entire space in the given reference frame, you can not take partial volumes within the interior of the spatial support of the tensor field, because this would not lead to a tensor for the corresponding spatial integral.

Does the relativity of simultaneity imply the impossibility of non local conservation laws?

It seems that by non-local conservation you mean that something disappears at one place and at the same time appears somewhere else. Then obviously relativity of simultaneity forbids them. Because what is simultaneous in one frame need not be simultaneous in another. So if a cat disappears here now and appears in Greece now (Feynman's example), for a moving observer it will disappear at some instant and appear at a later instant (for example).

Staff: Mentor

can I start from the integral ##Q^{\nu \rho \ldots}## and the assumption that it is a constant in time to prove that ##Q^{\nu \rho\ldots}## is a tensor

Yes, if you realize that the integral you wrote down has ##d^3 \vec{x}## in it, which means it requires picking a particular frame in which to do the integral. So the integral you wrote down, as it stands, is not relativistically invariant. To make it relativistically invariant, you have to do one of two things: (1) figure out a way to rewrite it so it has a relativistically invariant integrand, or (2) require that the integral as you wrote it must be valid in all inertial frames. Either way, you can then deduce the local differential form of the law.

Do you have a reference, where this is really done? That's then a really strong theorem, making non-local theories even less likely to work than it's apparent so far only by the absence of any working example.

Do you have a reference, where this is really done? That's then a really strong theorem, making non-local theories even less likely to work than it's apparent so far only by the absence of any working example.

The basic idea seems pretty trivial to me, and I think Feynman's argument is clear and convincing. However, the following may also be of interest:

Thanks, I'll have a look at the paper later, but I don't think that this is "pretty trivial". From the point of view of the representation theory of the proper orthochronous Poincare group, you start by looking for the irreducible ray representations of its covering group (which boils down to substitute the ##\mathrm{SO}(1,3)^{\uparrow}## by its covering group ##\mathrm{SL}(2,\mathbb{C})##)), which turn out to be all induced from unitary representations. In the next step from all these representations you choose a subset, which is governed by additional assumptions, among which the most important is locality, i.e., that you use field operators that transform under the representations in a local way like classical fields. These are specific mode decompositions. Together with microcausality and the boundedness of the Hamiltonian from below this leads to the very successful relativistic QFTs, used in the Standard Model, including the validity of the spin-statistics and the CPT theorem. But in this usual textbook treatment (see Weinberg, QT of Fields, Vol. 1 for a systematic treatment for massive and massless quanta of arbitrary spin) the locality in the one or the other form (Weinberg uses the linked-cluster theorem, which I think is also very convincing) is an additional assumption. So if there is a proof of its necessity, this cannot be that trivial.